case | sequence | recursive formula | OEIS
|
---|
R-T-V even RT-TV-VR even
| … 129, 26, 6, 3, 5, 13, 38 …
| an = 5an−1 − 6an−2 + an−3
| OEIS A198636
|
---|
R-T-V odd RT-TV-VR odd
| … 57, 11, 2, 1, 4, 16, 57 …
| Alternate terms of OEIS A096975
|
---|
The reverse of the recursive formula is:
αn = 6αn−1 − 5αn−2 + αn−3
which will always provide integer outputs for integer inputs. Thus it is not surprising that the R-T-V sequences succeed in yielding septrights for negative powers.
|
Sequences of identities involving powers of Q, S, U
|
---|
Q-S-U even | Q-S-U odd
|
---|
Q 0 | + S 0 | + U 0 | = | 3
| Q 2 | + S 2 | + U 2 | = | 7
| Q 4 | + S 4 | + U 4 | = | 21
| Q 6 | + S 6 | + U 6 | = | 70
| Q 8 | + S 8 | + U 8 | = | 245
| Q 10 | + S 10 | + U 10 | = | 882
| Q 12 | + S 12 | + U 12 | = | 3,234
|
|
Q 1 | + S 1 | − U 1 | = | √7
| Q 3 | + S 3 | − U 3 | = | 4 √7
| Q 5 | + S 5 | − U 5 | = | 14 √7
| Q 7 | + S 7 | − U 7 | = | 49 √7
| Q 9 | + S 9 | − U 9 | = | 175 √7
| Q11 | + S11 | − U11 | = | 637 √7
| Q13 | + S13 | − U13 | = | 2,352 √7
|
|
QS-SU-UQ even | QS-SU-UQ odd
|
---|
(QS) 0 | + (SU) 0 | + (UQ) 0 | = | 3
| (QS) 2 | + (SU) 2 | + (UQ) 2 | = | 14
| (QS) 4 | + (SU) 4 | + (UQ) 4 | = | 98
| (QS) 6 | + (SU) 6 | + (UQ) 6 | = | 833
| (QS) 8 | + (SU) 8 | + (UQ) 8 | = | 7,546
| (QS) 10 | + (SU) 10 | + (UQ) 10 | = | 69,629
| (QS) 12 | + (SU) 12 | + (UQ) 12 | = | 645,869
|
|
(QS) 1 | − (SU) 1 | − (UQ) 1 | = | 0
| (QS) 3 | − (SU) 3 | − (UQ) 3 | = | 21
| (QS) 5 | − (SU) 5 | − (UQ) 5 | = | 245
| (QS) 7 | − (SU) 7 | − (UQ) 7 | = | 2,401
| (QS) 9 | − (SU) 9 | − (UQ) 9 | = | 22,638
| (QS) 11 | − (SU) 11 | − (UQ) 11 | = | 211,288
| (QS) 13 | − (SU) 13 | − (UQ) 13 | = | 1,966,419
|
|
case | sequence | recursive formula | OEIS
|
---|
Q-S-U even
| 2, 2, 3, 7, 21, 70, 245 …
| bn = 7bn−1 − 14bn−2 + 7bn−3
| Multiply OEIS A122068 by 7
|
---|
Q-S-U odd
| 0, 1, 4, 14, 49, 175, 637 …
| OEIS A215493
|
---|
The reverse of the recursive formula above is:
βn = 2βn−1 − 1βn−2 + 1⁄7 βn−3
which involves division by 7, meaning that there is no guarantee that integer inputs will give rise to integer outputs. This helps explain why the Q-S-U sequences do not always give integer results for negative exponents. The same idea applies to the formula below, which when reversed will divide by 49.
|
QS-SU-UQ even
| 3, 14, 98, 833, 7546, 69629, 645869 …
| cn = 14cn−1 − 49cn−2 + 49cn−3
| Alternate terms of OEIS A275831
|
---|
QS-SU-UQ odd
| 0, 21, 245, 2401, 22638, 211288, 1966419
|
---|
qsu3 ÷ rtv3
|
---|
QV-ST-UR even — OEIS A108716
| QV-ST-UR odd — OEIS A215794
|
(Q÷V) 0 | + (S÷T) 0 | + (U÷R) 0 | = | 3
| (Q÷V) 2 | + (S÷T) 2 | + (U÷R) 2 | = | 21
| (Q÷V) 4 | + (S÷T) 4 | + (U÷R) 4 | = | 371
| (Q÷V) 6 | + (S÷T) 6 | + (U÷R) 6 | = | 7,077
| (Q÷V) 8 | + (S÷T) 8 | + (U÷R) 8 | = | 135,779
| (Q÷V) 10 | + (S÷T) 10 | + (U÷R) 10 | = | 2,606,261
| (Q÷V) 12 | + (S÷T) 12 | + (U÷R) 12 | = | 50,028,755
|
|
(Q÷V) 1 | − (S÷T) 1 | − (U÷R) 1 | = | √7
| (Q÷V) 3 | − (S÷T) 3 | − (U÷R) 3 | = | 31 √7
| (Q÷V) 5 | − (S÷T) 5 | − (U÷R) 5 | = | 609 √7
| (Q÷V) 7 | − (S÷T) 7 | − (U÷R) 7 | = | 11,711 √7
| (Q÷V) 9 | − (S÷T) 9 | − (U÷R) 9 | = | 224,833 √7
| (Q÷V) 11 | − (S÷T) 11 | − (U÷R) 11 | = | 4,315,871 √7
| (Q÷V) 13 | − (S÷T) 13 | − (U÷R) 13 | = | 82,846,113 √7
|
|
QR-SV-UT even
| QR-SV-UT odd
|
(Q÷R) 0 | + (S÷V) 0 | + (U÷T) 0 | = | 3
| (Q÷R) 2 | + (S÷V) 2 | + (U÷T) 2 | = | 14
| (Q÷R) 4 | + (S÷V) 4 | + (U÷T) 4 | = | 154
| (Q÷R) 6 | + (S÷V) 6 | + (U÷T) 6 | = | 1,883
| (Q÷R) 8 | + (S÷V) 8 | + (U÷T) 8 | = | 23,226
| (Q÷R) 10 | + (S÷V) 10 | + (U÷T) 10 | = | 286,699
| (Q÷R) 12 | + (S÷V) 12 | + (U÷T) 12 | = | 3,539,221
|
|
(Q÷R) 1 | + (S÷V) 1 | + (U÷T) 1 | = | 2 √7
| (Q÷R) 3 | + (S÷V) 3 | + (U÷T) 3 | = | 17 √7
| (Q÷R) 5 | + (S÷V) 5 | + (U÷T) 5 | = | 203 √7
| (Q÷R) 7 | + (S÷V) 7 | + (U÷T) 7 | = | 2,499 √7
| (Q÷R) 9 | + (S÷V) 9 | + (U÷T) 9 | = | 30,842 √7
| (Q÷R) 11 | + (S÷V) 11 | + (U÷T) 11 | = | 380,730 √7
| (Q÷R) 13 | + (S÷V) 13 | + (U÷T) 13 | = | 4,700,031 √7
|
|
At the time the present report was prepared, OEIS did not index any order-3 linear recurrences with those coefficients, even those that might have other initial values.